What is the binary equivalent of the decimal number 0.3125?

Question

Prepp Team · Accepted Answer

Understanding Decimal to Binary Conversion for Fractions

Converting a decimal number, especially a fractional one, to its binary equivalent involves a specific process. For fractional decimal numbers (numbers after the decimal point), we use the method of repeated multiplication by 2. Each time we multiply the fractional part by 2, the integer part of the result becomes the next digit in the binary fraction, reading from left to right after the binary point. We continue this process with the new fractional part until the fractional part becomes 0 or we reach the desired number of binary places.

Step-by-Step Conversion of 0.3125 to Binary

Let's convert the decimal number 0.3125 to its binary equivalent using the multiplication method.

Step 1: Multiply the fractional part (0.3125) by 2.

$0.3125 \times 2 = 0.625$

The integer part is 0. This is the first binary digit after the binary point. The new fractional part is 0.625.

Step 2: Multiply the new fractional part (0.625) by 2.

$0.625 \times 2 = 1.25$

The integer part is 1. This is the second binary digit. The new fractional part is 0.25.

Step 3: Multiply the new fractional part (0.25) by 2.

$0.25 \times 2 = 0.50$

The integer part is 0. This is the third binary digit. The new fractional part is 0.50.

Step 4: Multiply the new fractional part (0.50) by 2.

$0.50 \times 2 = 1.00$

The integer part is 1. This is the fourth binary digit. The new fractional part is 0.00.

Since the fractional part is now 0, we stop the process. The binary digits obtained from the integer parts (read from top to bottom) are 0, 1, 0, 1.

Resulting Binary Equivalent of 0.3125

Combining the integer part (which is 0 for 0.3125) and the fractional binary digits, we get the binary equivalent:

$0.3125_{10} = 0.0101_2$

Comparing with Options

Let's check the options provided:

Option 1: 0.0111
Option 2: 0.1010
Option 3: 0.0101
Option 4: 0.1101

Our calculated binary equivalent, 0.0101, matches Option 3.

Decimal Fraction	Multiply by 2	Result	Integer Part (Binary Digit)	New Fractional Part
0.3125	$0.3125 \times 2$	0.625	0	0.625
0.625	$0.625 \times 2$	1.25	1	0.25
0.25	$0.25 \times 2$	0.50	0	0.50
0.50	$0.50 \times 2$	1.00	1	0.00

Revision Table: Key Conversion Concepts

Concept	Description	Method for Fractional Decimal to Binary
Decimal Number System	Base-10 system, uses digits 0-9. Positional notation where each digit's value depends on its position and is a power of 10.	N/A
Binary Number System	Base-2 system, uses digits 0 and 1. Positional notation where each digit's value depends on its position and is a power of 2.	N/A
Fractional Conversion	Converting the part of a number after the decimal (or binary) point.	Repeatedly multiply the fractional part by 2. The integer part of the result is the next binary digit. Continue with the new fractional part.

Additional Information on Number Systems and Conversions

Understanding how to convert between different number systems like decimal (base-10) and binary (base-2) is fundamental in computer science and digital electronics. While fractional conversion uses repeated multiplication by the target base (2 for binary), integer conversion uses repeated division by the target base, taking the remainders as the digits (read from bottom to top).

For example, converting the integer part 3125 to binary would involve dividing 3125 by 2 repeatedly. However, for the number 0.3125, the integer part is 0, which remains 0 in binary.

Other common number systems include octal (base-8) and hexadecimal (base-16). Conversion methods similar to decimal to binary (repeated division for integers, repeated multiplication for fractions) can be applied, but using the target base instead of 2.